Obtain the expression for deflecting torque acting on the current carrying rectangular coil of a galvanometer in a uniform magnetic field. Why is a radial magnetic field employed in the moving coil galvanometer?

**Deflecting Torque on a Current-Carrying Rectangular Coil in a Uniform Magnetic Field:**

Consider a rectangular coil with `N` turns, having length `l` and breadth `b`. The coil is placed in a uniform magnetic field `B`. When a current `I` flows through the coil, the magnetic force acts on each side of the coil, resulting in a torque that causes the coil to rotate.

1. **Magnetic Force on a Single Side of the Coil:**
The magnetic force acting on each side of the coil is given by the formula:
F = B * I * l
where:
- `B` is the magnetic field,
- `I` is the current,
- `l` is the length of the side.

2. **Torque on the Coil:**
The torque on the coil due to the magnetic force is the product of the force and the perpendicular distance from the axis of rotation. For a rectangular coil with two opposite sides in the field, the torque for one side is given by:
τ = F * d
where `d` is the distance from the axis of rotation to the line of action of the force.

For the rectangular coil, the total torque is the sum of the torques acting on the two sides, which gives:
τ = 2 * (B * I * l * b)
where `b` is the breadth of the coil.

Therefore, the total torque acting on the rectangular coil is:
τ = N * B * I * A
where:
- `N` is the number of turns of the coil,
- `A = l * b` is the area of the rectangular coil.

Thus, the expression for the deflecting torque on the current-carrying rectangular coil is:
τ = N * B * I * A

---

**Why is a Radial Magnetic Field Employed in the Moving Coil Galvanometer?**

A radial magnetic field is employed in the moving coil galvanometer for the following reasons:

1. **Uniform Torque:**
In a radial magnetic field, the direction of the magnetic field is perpendicular to the plane of the coil at every point along the coil. This ensures that the force on each element of the coil is always perpendicular to the plane of the coil, resulting in a uniform torque. A uniform torque provides a more accurate and consistent deflection for the galvanometer.

2. **Linear Relationship Between Current and Deflection:**
In a radial field, the torque is directly proportional to the current flowing through the coil, which helps in achieving a linear relationship between the current and the deflection of the needle. This makes the scale of the galvanometer more uniform and easier to read.

3. **Minimizing Distortions:**
A radial field helps in minimizing distortions or non-uniformities in the magnetic field that could arise if the field were not radial. A non-uniform field could lead to uneven forces on different parts of the coil, resulting in inaccurate deflections.

In conclusion, the radial magnetic field ensures a consistent, linear, and uniform response of the galvanometer, making it more precise for measuring small currents.